Tagged mapping class groups: Auslander-Reiten translation

TL;DR

This paper introduces a geometric model called the tagged rotation to realize the AR-translation in generalized cluster categories derived from surfaces with marked points, extending previous results to punctured cases.

Contribution

It generalizes the geometric realization of AR-translation to surfaces with punctures, linking it to the tagged rotation and exploring its implications in derived categories and braid groups.

Findings

The tagged rotation models AR-translation for punctured surfaces.

The intersection of shifts and braid groups is empty except for certain polygonal surfaces.

Extension of Brüstle-Zhang's results to punctured cases.

Abstract

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface $\mathbf{S}$ with marked points and non-empty boundary, which generalizes Br\"{u}stle-Zhang's result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi-Yau derived category $\mathcal{D}(\Gamma_{\mathbf{S}})$ associated to the surface and the corresponding Seidel-Thomas braid group of $\mathcal{D}(\Gamma_{\mathbf{S}})$ is empty, unless $\mathbf{S}$ is a polygon with at most one puncture (i.e. of type A or D).